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Modeling Multibody Dynamic Systems With Uncertainties. Part I: Theoretical and Computational Aspects
Adrian Sandu*, Corina Sandu§, and Mehdi Ahmadian§
Virginia Polytechnic Institute and State University *Computer Science Department, [email protected]
§Mechanical Engineering Department, {csandu, ahmadian}@vt.edu
Abstract This study explores the use of generalized polynomial chaos theory for modeling complex nonlinear multibody dynamic systems in the presence of parametric and external uncertainty. The polynomial chaos framework has been chosen because it offers an efficient computational approach for the large, nonlinear multibody models of engineering systems of interest, where the number of uncertain parameters is relatively small, while the magnitude of uncertainties can be very large (e.g., vehicle-soil interaction). The proposed methodology allows the quantification of uncertainty distributions in both time and frequency domains, and enables the simulations of multibody systems to produce results with “error bars”.
The first part of this study presents the theoretical and computational aspects of the polynomial chaos methodology. Both unconstrained and constrained formulations of multibody dynamics are considered. Direct stochastic collocation is proposed as less expensive alternative to the traditional Galerkin approach. It is established that stochastic collocation is equivalent to a stochastic response surface approach. We show that multi-dimensional basis functions are constructed as tensor products of one-dimensional basis functions and discuss the treatment of polynomial and trigonometric nonlinearities. Parametric uncertainties are modeled by finite-support probability densities. Stochastic forcings are discretized using truncated Karhunen-Loeve expansions.
The companion paper “Modeling Multibody Dynamic Systems With Uncertainties. Part II: Numerical Applications” illustrates the use of the proposed methodology on a selected set of test problems. The overall conclusion is that despite its limitations, polynomial chaos is a powerful approach for the simulation of multibody systems with uncertainties. Keywords: multibody system, uncertainty, polynomial chaos, stochastic ODE, stochastic DAE.
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Introduction Practical mechanical systems often operate with some degree of uncertainty. The uncertainties can result from poorly known or variable parameters (e.g., variation in suspension stiffness and damping characteristics), from uncertain inputs (e.g., soil properties in vehicle-terrain interaction), or from rapidly changing forcings that can be best described in a stochastic framework (e.g., rough terrain profile). For realistic predictions of the system behavior and performance multibody dynamic models must account for these uncertainties.
Current methods used to formally assess uncertainties include Monte Carlo simulations, and linear and nonlinear approximations of the system response. Monte Carlo simulations are costly, and the accuracy of the estimated statistical properties improves with only the square root of the number of runs. Perturbation, statistical linearization, and nonlinear approximation methods compute several moments of the uncertainty distribution of the solution, but do not capture essential features of the nonlinear dynamics (e.g., as revealed by power spectral density).
This study applies the generalized polynomial chaos theory to formally assess the uncertainty in multibody dynamic systems. We investigate the computational aspects of incorporating various types of uncertainties into multibody dynamic system models, and illustrate the methodology presented on practical examples related to vehicle dynamics and mobility in off-road conditions. The approach used is to extend the model along the stochastic dimension to explicitly parameterize the uncertainty distribution. Polynomial chaos offers an efficient computational approach for the large, nonlinear multibody models of engineering systems of interest. For such systems, the number of uncertain parameters is relatively small, while the magnitude of uncertainties can be very large (e.g., vehicle-soil interaction).
The methods discussed in this study allow the quantification of uncertainties and enable the simulations of multibody systems to produce results with “error bars”, similar to the way the experimental results are often presented. Moreover, the proposed methodology allows the quantification of uncertainties in both time and frequency domains.
Polynomial chaos has been successfully applied in structural mechanics and in fluid mechanics studies. To our knowledge, this is the first application to multibody dynamics. It applies polynomial chaos to differential algebraic equations (DAEs) for constrained systems. It proposes direct statistical collocation as an alternative to the Galerkin method, and establishes its equivalence to a surface response approach. It presents a computationally efficient way to treat polynomial nonlinearities based on splitting the multi-dimensional integrals.
The study is organized in two parts. Part I (the current paper) presents the theoretical and computational aspects of the study. After providing a background on methods currently available to treat uncertainties in multibody dynamic systems, and the generalized polynomial chaos expansion, the paper discusses modeling of parametric and external sources of uncertainty in mechanical
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systems. Next, it introduces the basis functions used for the representation of the stochastic dimension. This is followed by presenting the stochastic ordinary differential equation formulation of multibody dynamics, and the stochastic differential algebraic formulation. The paper continues with the analysis of the uncertainty in model results in both time and frequency domains. Finally, the conclusions of this study are presented. The Legendre, Jacobi, and Hermite polynomials are given in the appendix. Part II is the complementary paper [1] which presents numerical results and discussions for representative case studies.
Background In this section we review the equations governing multibody dynamics, discuss the methods currently employed to treat uncertainties in such systems, and give an overview of the polynomial chaos approach. Polynomial chaos has been used extensively to model uncertainties in structural mechanics and in fluids, but to our knowledge it has not been previously applied to multibody dynamic simulations.
Formulations of Multibody Dynamic Equations The dynamics of a multibody system can be described in local or in global coordinates, in Cartesian or in generalized coordinates. The dynamics of an unconstrained mechanical system [2] can be described by a set of simultaneous fist order differential equations (ODE):
( ) FtttytypvytFvvy ≤≤=== 000 ,,);,,(, && (1) Here dy ℜ∈ are the generalized positions, dv ℜ∈ are the generalized
velocities, dv ℜ∈& are the generalized accelerations, and ep ℜ∈ is a vector of system parameters. The dot notation represents derivative with respect to time.
Using the same notation as in Eq. (1), one can represent a constrained mechanical system in Cartesian coordinates [3] by a system of index-3 differential algebraic equations (DAE):
⎪⎩
⎪⎨
⎧
=+=
=
0)(),,()(
yvytFvyM
vyTy
ψλψ&
&
(2)
We consider holonomic position constraints denoted by cd ℜ→ℜ:ψ ; dddM ×ℜ→ℜ: is the generalized mass matrix, dddF ℜ→ℜ×ℜ×ℜ: are the
external generalized forces and torques, cℜ∈λ represent the Lagrange multipliers, and λψ T
y represent the constraint forces. Partial derivatives are denoted by subscripts; e.g., yψ is the Jacobian of ψ with respect to y .
By differentiating the holonomic position constraints twice with respect to time, one obtains the constraint equations for velocity and acceleration:
),()(
0)(
vyvy
vy
y
y
τψ
ψ
=
=
& (3)
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Replacing the position constraints in Eq. (2) by the acceleration constraints from Eq. (3) leads to the index-1 DAE formulation, which is convenient for computational purposes [3]:
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡=
),(),,(
0)()()(
,vyvytFv
yyyM
vyy
Ty
τλψψ &
& (4)
The system in Eq. (4) is mathematically equivalent with the one given by Eq. (2), however a numerical scheme applied to Eq. (4) will lead to increasing errors in the position and velocity constraints [4]. To alleviate this drift-off, the numerical solution ( )nn vy ~,~ is projected at each time step onto the coordinate constraint manifold by solving the following nonlinear system for ny :
( )( ) ( ) ( ) 0,0~~~ ==+− nnTynnn yyyyyM ψηψ (5)
Similarly, the computed velocity is projected onto the velocity constraint manifold by solving the following linear system for nv :
( )( ) ( ) ( ) 0,0~ ==+− nnynTynnn vyyvvyM ψηψ (6)
In Eq. (6) cℜ∈η is another set of Lagrange multipliers. The projected values ( )nn vy , are the new numerical solution at nt .
Methods Available for Treating Uncertainties The traditional modeling approach assumes an ideal input with precisely defined parameters which determine the value of the output. When some parameter values or external forcings are not known, or cannot be accurately represented, the probabilistic framework is more appropriate. In this framework one models uncertain input parameters, and formulates the dynamic model to reflect the propagation of uncertainty in the output. Some of the commonly-used methods that are adapted for solving systems with uncertainties are now reviewed.
A very general approach is to solve the Fokker-Plank (FP) equation which governs the evolution of uncertainty distribution under system dynamics [5,6]. The method works in a high-dimensional probability space, and is not a practical computational tool for systems of interest.
Monte Carlo approach has been used extensively in dynamic models. An ensemble of runs is performed with each member using a different set of parameters drawn from the corresponding uncertainty distribution [7-9]. The statistical properties of the outputs are obtained from the ensemble of simulation results. The approach is computationally expensive as the estimation of the variance converges with the inverse square root of the number of runs. Related, more economical approaches are Latin Hypercube Sampling [10] and Bayesian Monte Carlo [11-12].
The perturbation (sensitivity) approach uses first and higher order sensitivity coefficients to derive low order moments of the simulation uncertainty [13-15]. This approach is useful when uncertainties are small, and behave like perturbations of the model [16,17].
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The Neumann series expansion of the governing stochastic operator is described in [18]. Application of this technique to large nonlinear systems is, however, difficult.
Response approximation methods build simplified models to capture the relationship between the uncertain inputs and uncertain outputs. Statistical linearization [19,20] and quadratization [21] produce models with the correct statistical moments. However, they do not provide any information on power spectral density of the response, which is important to analyzing uncertain mechanical systems. Finite order Volterra series [22] and nonlinear methods [23] were used to describe the input-output relation of nonlinear systems with stochastic excitations. These methods are difficult to apply to systems with multiple degrees of freedom. The deterministic equivalent model [24,25] and the stochastic response surface approach [26-29] use polynomial chaos representations of the inputs and outputs, and determine the coefficients of the output model through a collocation approach based on a small number of model runs.
Stochastic averaging [30-32] and non-Gaussian closure techniques [33,34] pursue information regarding the statistical moments of the results. Stochastic averaging has proven to be effective for deriving approximate solutions for weekly damped systems. Its application to nonlinear dynamic systems, which are often moderately to heavily damped, has proven difficult.
The approach employed by [35,36] in the context of finite elements for solid mechanics uses a spectral approximation the uncertainty which allows high order representations. The fundamental ideas stem from Wiener's homogeneous chaos theory [37]. Comprehensive work was done by Ghanem and co-workers for modeling uncertainty in other applications including nonlinear vibrations, fluids, porous media, etc. [21,35,36,38-43]. Karniadakis and co-workers introduced the concept of generalized polynomial chaos, and studied extensively its use to model uncertainties in fluids, including applications such as advection and diffusion, turbulence, and flow-structure interactions [44-55].
Keese presents a comprehensive review of recent methods for the numerical solution of stochastic partial differential equations [56].
Polynomial chaos offers a tractable computational approach for the large, nonlinear multibody models of engineering systems of interest. For such systems the number of uncertain parameters is relatively small, while the magnitude of uncertainties can be very large (e.g., vehicle-soil interaction).
Generalized Polynomial Chaos Expansion Polynomial chaoses [37,57] are generalizations of polynomials to the case where the independent variables are themselves measurable functions (in this paper random variables). The fundamental idea is that random processes of interest can be approximated (with arbitrary accuracy) by sums of orthogonal polynomial chaoses of random independent variables. These sums offer a computationally attractive approach to representing the state of physical systems operating under uncertainty.
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Second order random processes are processes with finite variance; from a physical point of view they have finite energy. A second order random process ( )θX , viewed as a function of the random event θ , ( )10 << θ , can be expanded in terms of orthogonal polynomial chaoses as [35,36]:
( )∑∞
=
=1
)()(j
jjcX θξφθ (7)
Here ( )nii
i ξξφ K1
are generalized Askey-Wiener polynomial chaoses of order )(in , in terms of the multi-dimensional random variable ( )
nii ξξξ K1
= . For Gaussian random variables the basis are Hermite polynomials, for uniformly distributed random variables the basis are Legendre polynomials, for beta distributed random variables the basis are Jacobi polynomials, and for gamma distributed random variables the basis are Laguerre polynomials [46.47]. The basis chaos polynomials form a complete orthogonal basis for the Hilbert space of square integrable random variables,
jifor0, ≠=ji φφ (8) This orthogonality relation holds with respect to the ensemble average
inner product, ( ) ( ) ( ) ξξξξ dwgfgf ∫=, (9)
Here )(ξw is the joint probability density of the random variables ξ . The series in Eq. (7) converges to any random process in 2L sense [57]. In practice, a truncated expansion is used,
( )∑=
=S
j
jjcX1
ξφ (10)
This means that we consider a finite number n of random variables ( )nξξξ K1= , and sum polynomials only up to a maximal order P . The total
number of terms )!!()!( PnPnS += increases rapidly with the number of stochastic parameters n and the order of the polynomial chaos P . Note that statistical bilinearization, for example, also constructs a system that is considerably larger than the original [22]. In Eq. (10), and for the remaining of this paper, we drop the explicit dependence of the random variable on the event θ .
Modeling Sources of Uncertainty in Mechanical Systems Dynamic systems are often affected by multiple sources of uncertainty. The main classes are parametric uncertainty and uncertain external excitations. In this section we discuss several approaches to model these uncertainties.
Parametric Uncertainty Uncertain parameters in mechanical systems take values between well defined bounds. Consequently, they cannot be accurately represented as normal random variables, since the Gaussian distribution has infinite support. We will focus on probability densities with finite support, namely the uniform and beta distributions.
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These distributions, defined on the finite interval ]1,1[−∈ξ , have the following probability density functions (PDF):
1. Uniform probability distribution:
( ) 21=ξw (11)
2. Beta probability distribution:
( ) ( )( ) ( ) ( ) ( ) ( ) ∫
∞ −−++ =Γ+−⋅
+Γ+Γ++Γ
=0
11 xwhere,11
1122 dtetba
baw txbaba ξξξ (12)
Here )(xΓ denotes the gamma function. The beta distribution has two parameters, a and b , which define the shape of the distribution, as illustrated in Fig. 1. For 0== ba the beta distribution reduces to the uniform probability distribution.
−1 −0.5 0 0.5 10
0.3
0.6
0.9
1.2
ξ
uniform
−1 −0.5 0 0.5 10
0.3
0.6
0.9
1.2
ξ
beta(1,1)
−1 −0.5 0 0.5 10
0.3
0.6
0.9
1.2
ξ
beta(3,1)
(a) Uniform distribution (b) Beta distribution (c) Beta distribution
Figure 1. Examples of distributions with finite support
Parametric Representation of Stochastic Forcing An important aspect in the study of vehicle behavior over rough terrain is modeling the uncertainty in the terrain profile. More generally, an important aspect in the study of mechanical systems is the representation of uncertain external forcings. In this section, the external forcing depends on x (space) instead of t (time) to intuitively represent terrain variation. The discussion, however, is general and can be directly applied to any time dependent stochastic forcing function.
The terrain profile is considered a random process ( )xz defined over the spatial domain Dx∈ . The random terrain height can be represented as a mean height ( )xz plus a sum of deterministic shapes ( )xgk multiplied by random amplitudes kξ ,
( ) ( ) ( )∑=
+=n
kkk xgxzxz
1
ξ (13)
The shape functions are linearly independent, e.g., can be chosen from an orthonormal base of the set of square integrable functions [ ]( )DL ,02 .If the
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random amplitudes kξ are assumed to be independent identically distributed random variables, with zero mean and variances σk, the forcing covariance function is:
( ) ( )∑=
=−−=n
kkkk xgxgxzxzxzxzxxR
121
2221121 )()(),()(),( σ (14)
Similarly, if the terrain profile ( )θ,xz has known covariance ( )21, xxR , then the random profile can be represented by the Karhunen-Loeve (KL) expansion [35,36,46-52]:
( ) ( ) ( )∑∞
=
+=1k
kkk xgxzxz ξλ (15)
Here kξ is an independent set of random variables of mean 0 and variance 1. The shape functions ( )xgk are the eigenvectors, and kλ the eigenvalues of the covariance function:
( ) ( ) ( )∫ =D kkk xgdxxgxxR 12221 , λ (16)
A practical representation of the stochastic process is obtained by truncating the KL series in Eq. (14) to n terms, based on the relative magnitude of the eigenvalues, such that 11 λλ <<+n .
The deviation of the terrain surface from the mean profile )()( xzxz − is considered a wide-sense-stationary (WSS) process. A WSS random process has a constant mean and a covariance that depends only on the distance between two points:
( ) ( )212211 R)()(,)()(and0)( xxxzxzxzxzxzxz −=−−=− (17) The power density spectrum of a stationary random process, i.e., the
power density of ]))(),([( 2xzxzE −θ , quantifies the frequencies on which the process fluctuates. The spectrum of a stochastic terrain profile quantifies the roughness of the surface.
The power spectrum of a WSS random process is the Fourier transform of its covariance function (Wiener-Khinchine theorem [35,36]),
( ) ( )∫ −=D
xi dxexRR ωωˆ (18)
The characteristic frequencies of the stochastic forcing are given by: ( ) kkR λω =ˆ (19)
where kλ are the eigenvalues in Eq. (16). Using the power spectrum approach, the stochastic forcing (terrain) can
be modeled as follows: • Measure the power spectrum of the forcing (terrain) variance • Calculate the covariance function )(τR as the inverse Fourier
transform of forcing (terrain) power spectrum, and • Calculate the largest eigenvalues of the covariance and the
corresponding eigenvectors, and use them to build the Karhunen-Loeve stochastic representation of the forcing (terrain surface).
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Basis Functions To construct the polynomial chaos approximation, a set of orthogonal polynomial basis functions is defined with respect to the probability density function. For the one-dimensional uniform distribution the basis functions are the Legendre polynomials:
( ) ( )∫+
−≠=
1
1for0
21 jidLL ji ξξξ (20)
and for the one-dimensional beta distribution, the Jacobi polynomials:
( ) ( ) ( )( ) ( )( ) ( )∫
+
− ++ ≠=+−+Γ+Γ
++Γ1
1 1 for011112
2 jidba
baJJ babaji ξξξξξ (21)
The Jacobi and Legendre polynomials up to order ten are given in the appendix. We take the beta distribution with 1== ba for a symmetric probability density function.
In this paper we restrict the discussion to multiple independent random variables nξξ ,,1 K with a joint distribution of probability
( ) ( )( ) ( )( ) ( )( )nnn wwww ξξξξξ KK 2
21
11 = (22)
The basis functions are orthogonal in the joint probability space: ( ) ( ) ( ) jiforddw nnn
jn
iji ≠== ∫ ∫+
−
+
−0,
1
1
1
1 1111 ξξξξξξφξξφφφ KKKKK (23)
A multi-dimensional orthogonal basis is constructed as follows. Let { } 0
)(≥k
jkP be the family of one-dimensional polynomials orthogonal with respect to
the density ( )jw , and consider basis functions defined by tensor products of such polynomials:
( ) ( )( ) ( )( ) ( )( )nniiin
inPPP ξξξξξφ KK 1
21
11 21
= (24) The choice presented in Eq. (24) enables the evaluation of n -dimensional
scalar products from n separate one-dimensional scalar products. The orthogonality condition from Eq. (23) becomes:
( ) ( ) ( ) ( ) ( ) ( ) nkjidwPP kkkkk
kkj
n
kk
ki
jikk
≤≤≠≠=∏ ∫=
+
−1allforifonly0,
1
1
1ξξξξφφ (25)
Therefore, the tensor products of orthogonal polynomials in Eq. (24) form an orthogonal set in the n -dimensional space of functionals. A generalization of the Cameron and Martin theorem [57] was employed in [46] to conclude that each type of Askey chaos converges to any L2 functional in the L2 sense in the corresponding Hilbert space. In particular, multi-dimensional Legendre and Jacobi polynomial chaoses form a complete orthogonal basis of the space of n -dimensional square integrable random variables.
In this paper, for simplicity, we consider that all variables nξξ ,,1 K are identically distributed, i.e., )()2()1( nwww ≡≡≡ K , although the more general case with independent variables drawn from different distributions can be treated similarly. Consequently, in what follows, the basis functions are tensor products of same-family polynomials.
In [35,36,59] basis functions given by Eq. (24) of multi-dimensional order up to P are considered. The total number of basis functions is:
10
!!)!(0 1 PnPnSPii n ⋅
+=⇒≤++≤ K (26)
This number increases quickly with the number of independent uncertain variables n and the order of the polynomial chaos P , as shown in Table 1.
Table 1. Number of polynomial chaos coefficients S
Another approach is to choose tensor products of one-dimensional polynomials up to order P , in this case the dimension of the stochastic space is
11 0,,0 +=⇒≤≤≤≤ P
n nSPiPi L (27) This setting is more natural for the collocation approach, as discussed below.
Stochastic ODE Formulation Consider the multibody dynamic system in the ODE formulation given by Eq. (1) with uncertain parameters. The uncertain parameters are (functional of) random variables and can be represented using the polynomial chaos expansion as:
ekppS
i
iikk ≤≤= ∑
=
1,)(1
ξφ (28)
The state variables of the multibody dynamic system are also functionals of the random variables that describe the sources of uncertainty, and are represented as:
( ) dktvtvtytyS
i
iikk
S
i
iikk ≤≤== ∑∑
==
1,)()(,)()()(11
ξφξφ (29)
We use subscripts to denote the components along the deterministic (system) dimension, and superscripts to denote components along the stochastic dimension. The superscript-only notation will be used to represent the vector of stochastic coefficients,
[ ] Siyyy Tid
ii ≤≤= 1allfor1 L (30) Inserting Eq. (30) into Eq. (1) leads to:
SjdkpvytFvvyS
m
mmS
m
mmS
m
mmk
jS
j
jk
ik
ik ≤≤≤≤⎟
⎠
⎞⎜⎝
⎛== ∑∑∑∑
====
1,1for,;,,,1111
φφφφ&& (31)
The equations for the time evolution of the spectral polynomial chaos coefficients )(tyik and )(tvik can be derived in either the Galerkin or collocation frameworks, as explained below.
Galerkin Approach
Equation (31) is projected onto { }Sspan φφ ,,1 K . Specifically, we take the ensemble average scalar product of Eq. (31) with each basis function iφ .
P=1 P=2 P=3 n=5 6 21 56 n=10 11 66 286 n=20 21 231 1,771
11
Considering the orthogonality relations, this procedure leads to the following (component-wise) model:
( ) ( ) Sidkyty
pvytFv
vy
ik
ik
iS
m
mmS
m
mmS
m
mmk
iiik
ik
ik
≤≤≤≤=
⎟⎠
⎞⎜⎝
⎛=
=
∑∑∑===
1,1for,
,;,,,
00
111φφφφφφ&
&
(32)
The model from Eq. (32) describes the time evolution of the polynomial chaos coefficients )(tyik and )(tvik . It is S times larger than the original model (1), and coupled through the nonlinear terms. Note that the differential equations in Eq. (32) for polynomial chaos coefficients are, by themselves, deterministic, and can be solved using traditional time stepping algorithms. The computational challenges stem from the dimensionality of this system, and from the nonlinear coupling terms which require the evaluation of multiple integral terms.
In the simulation of mechanical systems two types of nonlinearities are typically encountered: polynomial and trigonometric. For power nonlinearities
myyf =)( the stochastic Galerkin formulation leads to integrals of the form
∑∑ ∑∑= = ==
=⎟⎠
⎞⎜⎝
⎛ S
i
S
i
S
i
jiiiijmS
i
ii
m
mmyyy1 1 11 1 2
11 ,, φφφφφ KKK (33)
This expression involves n -dimensional integrals of 1+m products of basis functions. Since each basis function given by Eq. (24) is a tensor product of one-dimensional chaos polynomials, the n -dimensional integration reduces to computing n one-dimensional integrals,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )11
)()()(1
)(1 ,,1
1 ∏∏==
=⇒=n
rr
rrjr
rrir
rrin
jiin
rr
rrin
i PPPPn
m
k
k ξξξφφφξξξφ KKK (34)
Each one-dimensional integral can be evaluated using a Gaussian numerical quadrature (Gauss-Legendre for uniform distribution, and Gauss-Jacobi for beta distribution). The products in Eq. (34) can be pre-computed and used throughout the integration of Eq. (32).
For trigonometric nonlinearities, however, the multi-dimensional ensemble inner products cannot be split into products of one-dimensional integrals. Instead, these inner products have to be evaluated by a multi-dimensional quadrature with nodes qµµ K1 and weights qρρ K1 . The quadrature nodes are n -
dimensional vectors, )( 1ln
ll µµµ K= . The inner products are approximated as:
( ) ( ) ( )ljS
i
liiS
i
liiq
ll
jS
i
iiS
i
ii vytfvytf µφµφµφρφφφ ⋅⎟⎠
⎞⎜⎝
⎛⋅≈⎟
⎠
⎞⎜⎝
⎛ ∑∑∑∑∑===== 11111
,,,,, (35)
One possible choice is the Smolyak quadrature formulas [56]. To avoid multi-dimensional integrals we can expand trigonometric
functions in Taylor series
( )( )∑∞
=
−=0
00)()(
m
mmm xxxfaxf (36)
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For arguments represented by a polynomial chaos expansion, the Taylor becomes
( )∑ ∑ ∑∑∞
= = ==
==−0 1 1
0)(
10
1
11)(,m
S
i
S
i
iiiimm
S
i
ii
m
mmxxxfaxfxxx φφφ KKL (37)
Then the inner products can be expressed in a series
( )
( )∑ ∑ ∑
∑ ∑ ∑∑∞
= = =
∞
= = ==
=
=⎟⎠
⎞⎜⎝
⎛+
0 1 1,0
)(
0 1 10
)(
10
1
1
1
1
11 ,,
m
S
i
S
i
iijii
mm
m
S
i
S
i
jiiiimm
jS
i
ii
m
m
m
m
mm
xxExfa
xxxfaxxf
KL
KKL
L
φφφφφ (38)
where the products E can be computed as in Eq. (34). Other approximations of trigonometric nonlinearities (e.g., Pade) can also be employed.
Collocation Approach The collocation approach is motivated by the pseudo-spectral methods [58]. A variant of stochastic collocation for uncertainty analysis was proposed in [59] in the context of Gaussian uncertainties with Hermite polynomial chaos. The technique has been applied successfully in the study of compressible flows with uncertainty [60].
In order to derive evolution equations for the stochastic coefficients )(tyi we impose that Eq. (31) holds at a given set of collocation vectors:
[ ] SiTid
ii ≤≤= 1allfor1 µµµ L (39) This leads to:
( ) ( ) ( ) ( ) SipvytFvvyS
m
immS
m
immS
m
immijS
j
jii ≤≤⎟⎠
⎞⎜⎝
⎛== ∑∑∑∑====
1,;,,,1111
µφµφµφµφ&& (40)
Consider the matrix A of basis function values at the collocation points: ( ) ( ) SiSjAA ij
jiji ≤≤≤≤== 1,1,, ,, µφA (41) The collocation points have to be chosen such that A is nonsingular. Then:
( ) ∑∑==
=S
j
jji
S
j
ijj yAy1
,1
ξφ (42)
Equation (40) becomes:
SipAvAyAtFvAvyS
m
mmi
S
m
mmi
S
m
mmi
S
j
jji
ii ≤≤⎟⎠
⎞⎜⎝
⎛== ∑∑∑∑====
1,;,,,1
,1
,1
,1
, && (43)
Denote the collocation points in the random system state space by:
( ) ( ) ( ) ( ) ∑∑∑===
===S
j
jji
iS
j
jji
iS
j
jji
i pAPtvAtVtyAtY1
,1
,1
, ,, (44)
With this notation, the collocation system (43) can be written as: ( ) SiPVYtFVVY iiiiii ≤≤== 1,;,,, && (45)
Equation (45) shows that the direct collocation approach reduces to S independent solutions of the deterministic system (1). At 0=t , Eq. (44) gives the initial conditions for Eq. (45). Equation (45) shows that collocation is, in fact, a
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response surface approach method. In the response surface approach [24, 26-29] the states of the system at the initial and final times are given finite dimensional approximations shown in Eq. (10).
After integration, the stochastic solution coefficients are recovered using:
( ) ( ) ( ) ( ) ( ) ( )∑∑=
−
=
− ==S
j
jji
iS
j
jji
i TVTvTYTy1
,1
1,
1 , AA (46)
The collocation approach requires S independent runs, each using a different value for the random variables jµξ = . The coefficients of the polynomial chaos expansion at the final time are recovered using Eq. (46).
We now discuss the choice of collocation points. Let qγγ ,,1 K in ]1,1[− be the roots of the thq order one-dimensional polynomial from the family used in the construction of basis functions (Legendre or Jacobi). Since the basis functions iφ are tensor products, the n -dimensional collocation vectors are chosen to have each component equal to one of these points:
[ ] [ ] SjnjjTid
ii ≤≤== 1for11 γγµµµ LL (47)
There are qn possible collocation vectors. The dimension of the stochastic space with n independent sources of uncertainty and with chaos polynomials of order up to P is smaller than the number of possible collocation points, qnS < . One has to choose a subset of the vectors of form given by Eq. (47) as discussed in [24,28,29].The alternative approach is to construct the basis functions as tensor products of one-dimensional polynomials up to order 1−= qP , as given by Eq. (27). In this case 1+= PnS and the number of collocation points equals the base size.
Note on the Relation between Collocation and Galerkin Methods Consider the stochastic system in Eq. (37) obtained through collocation. Consider the Gaussian multi-dimensional quadrature with nodes qµµ K1 and weights
qρρ K1 , and use the quadrature nodes as collocation points. Multiply Eq. (39) by
)( iki µφρ and sum over i to obtain:
( ) ( ) ( ) ( ) ( ) ( )ikS
m
immS
m
immS
m
immS
ii
S
i
ikiji
S
j
j pvytFv µφµφµφµφρµφµφρ ⎟⎠
⎞⎜⎝
⎛= ∑∑∑∑∑∑
====== 111111;,,& (48)
Denote by { }⋅⋅, the ensemble average evaluated by numerical integration with the above choice of quadrature,
( ) ( ) ( ) ( ) ( ) { }∫ ∫ ∑+
−
+
−=
=≈=1
1
1
11
,, gfgfdwgfgf jjS
jj µµρξξξξL (49)
Equation (48) can be written as:
{ }⎭⎬⎫
⎩⎨⎧
⎟⎠
⎞⎜⎝
⎛= ∑∑∑∑
====
kS
m
mmS
m
mmS
m
mmS
i
kii pvytFv φφφφφφ ,;,,,1111
& (47)
14
If the quadrature method is sufficiently accurate to preserve the orthogonality of basis functions, then the left hand side sum contains only one nonzero term, and
{ } SkpvytFv kS
m
mmS
m
mmS
m
mmkkk ≤≤⎭⎬⎫
⎩⎨⎧
⎟⎠
⎞⎜⎝
⎛= ∑∑∑
===
1,,;,,,111
φφφφφφ& (48)
By comparing Eq. (51) and Eq. (32), we see that collocation can be regarded as the Galerkin method, with the ensemble averages evaluated numerically with a specific choice of numerical quadrature.
We first consider the one-dimensional case, where 1+= PS . The nodes of the Gaussian quadrature are the roots of the order S orthogonal polynomial with respect to the given probability density. For uniform distribution, one chooses the Gauss-Legendre, and for beta distribution the Gauss-Jacobi points. The quadrature nodes and weights (e.g., for uniform distribution) are:
( ) ( )( ) ( ) Sjdxxw
xLxxLL
SjS
jj
S ≤≤⋅−
== ∫+
−1,)(,0
1
1 'µρµ (52)
Gaussian quadrature integrates exactly polynomials up to order 12 −S , and the orthogonality relations are preserved, therefore (51) holds.
Replacing the exact integrals in Eq. (32) by the numerical quadrature formulas in Eq. (52) adds a numerical error term to the right hand side of the equation (left hand side is integrated exactly). For a smooth function f [61]:
( ) ( ) ( ) ( ) ( )∫ ∑+
−=
−∈⋅
+=1
1
)2(2
1]1,1[,
!2,
ηηµρ S
S
SSjS
jj f
SALL
fdxxwxf (53)
Here SA is the highest order coefficient in )(xLS . The quadrature truncation error from Eq. (53) decreases rapidly with increasing S and does not affect the spectral convergence rate of the Galerkin solution.
The multi-dimensional case requires the application of a tensor product Gaussian quadrature formula. For the basis functions from Eq. (27), the collocation points are exactly the quadrature nodes, and the conclusions of the one-dimensional analysis can be directly applied. For the basis functions from Eq. (26), the collocation points are a subset of the Gaussian nodes, and the numerical orthogonality relation leading to the form in Eq. (51) may not hold.
Note on the Relation between Direct and Stochastic Collocation The stochastic collocation algorithm of Mathelin et. al. [61,62] uses the cumulative distribution function (CDF) of a random variable to map its distribution to the interval ]1,0[ :
( ) [ ] ( )αξξξα 1,Pr −=≤== kkk y ΥΥ (54) A set of collocation points iα are chosen in the interval ]1,0[ (for example,
the Gauss-Legendre points). Each component of the random solution at a given time t is expressed in terms of the variable α
( ) ( ) ( )∑=
− ==S
i
iikkkk hyyy
1
1 ~)( ααξ Υ (55)
15
where the basis ih are the Lagrange interpolation polynomials based on the interpolation grid iα . The stochastic equation obtained by inserting Eq. (55) in Eq. (1) is formulated in terms of the stochastic variable α , and in [61,62] is solved using a Galerkin formulation in the α space and evaluating the integrals by quadrature. Imposing directly that the stochastic equation holds at the collocation points iα leads to
( ) SjdkpvytFvvy jjjk
jk
jk
jk ≤≤≤≤== 1,1for,~;~,~,~,~~ && (56)
Formally, the corresponding collocation points in the stochastic variable space are obtained by inverting the CDF of each random variable,
( ) ( ) ( ) ( ) Sjdkjkjv
kjkjy
k ≤≤≤≤== −− 1,1,, 11 αµαµ VΥ (57) Note that Eq. (55) can be related to the polynomial chaos expansion as
follows
( ) ( )( ) ( ) ( )∑∑∑∑====
=⇒==S
i
jyk
iik
jk
S
i
iik
S
ik
iik
S
i
iik yyyhyhy
1111)(~~~ µφξφξα Y (58)
A different set of collocation points in the stochastic variable space ξ is (formally) used for each system component. These collocation points are recalculated for each time step (time interval ],[ ttt ∆+ ). Eq. (58) is general and shows that the stochastic collocation method of [61,62] is not equivalent to the response surface method.
The method requires the numerical approximation of the CDF and its inverse for each system state variable. In [61,62] this is accomplished by numerical interpolations between the ξ and the α spaces. For multiple sources of uncertainty the inversion of CDF in Eq. (54) leads to an )1( −n -dimensional manifold in the random variable space; the collocation points need to be chosen along this manifold. It is not clear how to extend this approach to multiple sources of uncertainty.
Stochastic DAE Formulation We now consider multibody systems in the DAE formulation given by Eq. (2). To construct the corresponding stochastic system that describes the evolution of polynomial chaos coefficients, insert Eq. (30) into Eq. (2) to obtain:
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
=⎟⎠
⎞⎜⎝
⎛
⋅⎟⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛=⋅⎟
⎠
⎞⎜⎝
⎛=
∑
∑ ∑ ∑∑∑∑
=
= = ====
0
,,
1
1 1 1111S
m
mm
S
j
S
j
jjS
m
mmTy
S
m
mmS
m
mmjjS
m
mm
ii
y
yvytFvyM
vy
φψ
φλφψφφφφ &
&
(59)
The stochastic formulation will need to take the algebraic constraints into account. The stochastic evolution equations can be formulated using either the Galerkin or the collocation approach.
16
Galerkin Approach
In the Galerkin approach Eq. (59) is projected onto { }Sspan φφ ,,1 L . Taking the ensemble average scalar product of Eq. (59) with each basis function iφ and considering the orthogonality relations leads to the following stochastic system:
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
≤≤=⎟⎠
⎞⎜⎝
⎛
⎟⎠
⎞⎜⎝
⎛⋅+
⎟⎠
⎞⎜⎝
⎛=⎟
⎠
⎞⎜⎝
⎛⋅
=
∑
∑ ∑
∑ ∑∑∑
=
= =
= ===
Six
y
vytFyMv
vy
iS
m
mm
S
j
ijS
m
mmTy
j
S
j
iS
m
mmS
m
mmijS
m
mmj
ii
1for,0,
,
,,,,
1
1 1
1 111
φφψ
φφφψλ
φφφφφφ&
&
(60)
Equation (60) is an index 3 DAE for the stochastic coefficients with dS ⋅2 differential and cS ⋅ algebraic equations. The computational challenges
stem from the dimension and from the nonlinear scalar products (other than polynomial) which are expensive to calculate.
Collocation Approach In the collocation approach we impose the system given by Eq. (59) to hold exactly at the set of collocation points Sµµ K1 . With the notations of Eq. (41) and Eq. (44), this approach leads to the following formulation of the stochastic system:
( ) ( ) ( )( )⎪
⎩
⎪⎨
⎧
≤≤=
Λ⋅+=⋅=
SiY
YTyVYtFVYMVY
i
iiiiii
ii
1for,0
,,
ψ
ψ&
&
(61)
The index 1 formulation is: ( ) ( )( )
( )( ) ⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
Λ⎥⎥⎦
⎤
⎢⎢⎣
⎡=
ii
ii
i
i
iy
iTy
iii
VYVYtFV
YYYM
VY,,,
0 τψψ &
& (62)
The projection on the position manifold reads: ( ) ( ) ( ) ( ) SiYYYYYM iiT
yiii ≤≤==+−⋅ 1,0,0~~~ ψψ (63)
The projection on the velocity manifold is: ( ) ( )( )
( ) SiVYMVY
YYM ii
i
i
iy
iTy
i
≤≤⎥⎦
⎤⎢⎣
⎡ ⋅=⎥
⎦
⎤⎢⎣
⎡
Ν⎥⎥⎦
⎤
⎢⎢⎣
⎡1,
0
~
0ψψ (64)
Direct collocation reduces to the response surface approach, where the DAE is integrated independently S times with different initial values. The method is attractive due to its very simple implementation.
Note on the Formulation of Algebraic Constraints By solving the algebraic position constraint the state can be locally partitioned into dependent )( depy and independent )( indy coordinates.
17
( ) ( )inddep yhyy =⇔= 0ψ (65) The implicit and explicit formulations of the algebraic constraints given by
Eq. (65) are mathematically equivalent for the deterministic system. In the stochastic formulation:
⎟⎠
⎞⎜⎝
⎛= ∑∑
==
S
m
mmind
S
j
jjdep yhy
11φφ (66)
The uncertainties in the dependent and independent variables are correlated through the algebraic constraints. One would like to have the implicit and explicit formulations of the constraints equivalent in the stochastic formulation as well.
In the Galerkin formulation, the constraints are projected on the subspace of interest to hold. The implicit constraint formulation in Eq. (60) is, in general, different than the one obtained by projecting the explicit constraints of Eq. (66):
Siyhy iS
m
mmindii
idep ≤≤⎟
⎠
⎞⎜⎝
⎛= ∑
=
1,,,1
1φφ
φφ (67)
The implicit Eq. (61) and explicit Eq. (67) formulations of the algebraic constraints are equivalent in the stochastic collocation approach:
( ) SiYhYyAhyA iind
idep
S
j
S
m
mindmi
jdepji ≤≤=⇔⎟
⎠
⎞⎜⎝
⎛=∑ ∑
= =
1,1 1
,, (68)
Uncertainty in Results
Time Domain The statistics of any order for the output uncertainty can be derived from the polynomial chaos representation. The average of the thk model component is given by the th0 order term in the stochastic expansion,
( ) ( ) ( )tytyty kkk1== (69)
In general the mean is different than the model prediction using mean values for the uncertain parameters and forcing. The covariance matrix of the model state at any time is obtained as:
( ) ( ) ( ) ( ) ( )∑=
⟩⟨==S
i
iiim
ikmkmk tytytytytR
1, ,, φφ (70)
Using these measures, the model output can be visualized with an “error bar” representation of the uncertainty.
Frequency Domain The information in the frequency domain is essential for a thorough understanding of the dynamic system behavior. Nonlinear response phenomena in multibody dynamic systems include non-Gaussian response, multiple resonance frequencies, and high-energy low-frequency spectral content. The power spectrum of the deterministic system response is obtained by
( ) ( )ωyty Fourier ˆ⎯⎯ →⎯ (71)
18
A representation of the uncertainty in the power spectrum is obtained by a Fourier transform of the time series of each polynomial chaos coefficient,
( ) ( ) ( ) ( ) ( ) ( )ξφωξφω iS
i
iFourierS
i
iiiFourieri ytyyty ∑∑==
⎯⎯ →⎯⇒⎯⎯ →⎯11
ˆˆ (72)
The statistics and the probability density of the power spectral density of the model predictions can be derived from Eq. (79).
Conclusions This paper investigates the polynomial chaos methodology for the simulation of multibody systems with uncertainties. Uncertainties result from poorly known or variable parameters (e.g., variation in suspension stiffness and damping characteristics), from uncertain inputs (e.g., soil properties in vehicle-terrain interaction), or from rapidly changing forcings that can be best described in a stochastic framework (e.g., rough terrain profile).
Polynomial chaos expansion is used to discretize random processes by extending the solution along the stochastic dimension. The polynomial chaos basis functions are constructed as tensor products of 1-D orthogonal polynomials. This approach is chosen since multibody models are large and highly nonlinear for engineering systems of interest, the number of uncertain parameters is relatively small, while the magnitude of uncertainties can be very large (e.g., vehicle-soil interaction). In addition, the approach allows for the computation of the uncertainty in model outputs in both the time and frequency domains.
Multibody dynamic systems are considered in both the ordinary differential equation (ODE) and in differential algebraic equation (DAE) formulations. The DAE model appears in the simulation of constrained multibody systems. The construction of the evolution equations for the stochastic coefficients is discussed in both the Galerkin and collocation frameworks. The Galerkin approach to formulating the stochastic ODE is well established in the literature. This approach is extended in this study to formulate the stochastic DAEs. The direct stochastic collocation approach proposed here is motivated by pseudo-spectral methods. It is shown that direct collocation is equivalent to the construction of a stochastic response surface. Stochastic collocation is applied to both ODE and DAE formulations. A complete theory for the convergence of the direct collocation approach is not available at this time.
Multibody dynamic systems typically display two types of nonlinearities, polynomial and trigonometric. Ensemble averages need in the Galerkin approach lead to the evaluation of multi-dimensional integrals. It is shown that the tensor product nature of basis functions makes the evaluation of ensemble averages computationally efficient for polynomial nonlinearities. However, for trigonometric nonlinearities multi-dimensional integration is required, and this can be accomplished using numerical quadrature rules. Stochastic collocation only requires the evaluation of functions at different collocation points. It can be easily applied to any type of nonlinearities and uses only the deterministic simulation code.
19
This study considers uncertainties from both parametric and uncertain external forcing sources. The widely-used Gaussian distribution has infinite support and is not well suited to model uncertainties in mechanical systems, where uncertain parameters very between well defined bounds. We model parametric uncertainties using uniform and beta distributions. External stochastic forcings are modeled by truncated Karhunen-Loeve decompositions. We propose to calculate the covariance function of the stochastic process from the power spectrum of the forcing variance, which can be determined experimentally.
In the complementary paper “Modeling Multibody Dynamic Systems with Uncertainties. Part II: Numerical Applications” [1], we present several applications of the computational tools discussed here.
Acknowledgements The work of A. Sandu was supported in part by NSF through the awards CAREER ACI-0093139 and ITR AP&IM 0205198. The work of C. Sandu was supported in part by the AdvanceVT faculty development grant 477201 from Virginia Tech’ NSF ADVANCE Award 0244916.
Appendix. Legendre, Jacobi and Hermite Basis Functions The normalized Legendre basis functions, up to order 10, for 1 variable, are given by equations (A.1).
( )2
10 =xL ; ( ) xxL
23
1 = ; ( ) ( )22 31
225 xxL +−= ; ( ) ( )2
3 53227 xxxL +−=
( ) ( )424 35303
283 xxxL +−= ; ( ) ( )42
5 63701528
11 xxxxL +−= ;
( ) ( )6426 2313151055
21613 xxxxL +−+−= ; ( ) ( )642
7 42969331535216
15 xxxxL +−+−=
( ) ( )86428 6435120126930126935
212817 xxxxxL +−+−= (A.1)
( ) ( )86429 12155257401801846209135
212819 xxxxxxL +−+−=
( ) ( )4618910939590090300303465632256
21 864210 +−+−+−= xxxxxL
The normalized Jacobi polynomial basis on [-1,1] with a=1, b=1, up to order 10, for one variable, are given by equations (A.2).
( )23
0 =xJ ; ( ) xxJ215
1 = ; ( ) ( )22 51
6473 xxJ +−= ; ( ) ( )2
3 732453 xxxJ +−=
20
( ) ( )424 21141
516335 xxxJ +−= ; ( ) ( )42
5 33305316
913 xxxxJ +−=
( ) ( )6426 4294951355
764157 xxxxJ +−+−=
( ) ( )6427 715100138535
64153 xxxxxJ +−+−= (A.2)
( ) ( )86428 24314004200230875
768959 xxxxxJ +−+−=
( ) ( )86429 419979564914109263
52562315 xxxxxxJ +−+−=
( ) ( )10864210 29393629854641013650136521
225126911 xxxxxxJ +−+−+−=
The normalized Hermite polynomials are given by equations (A.3)
( ) 10 =xH ; ( ) xxH =1 ; ( ) ( )12
1 22 −= xxH ; ( ) ( )xxxH 3
61 3
3 −=
( ) ( )3662
1 244 +−= xxxH ; ( ) ( )xxxxH 1510
3021 35
5 +−=
( ) ( )154515512
1 2466 −+−= xxxxH ; ( ) ( )xxxxxH 10510521
35121 357
7 −+−=
( ) ( )105420210287024
1 24688 +−+−= xxxxxH (A.3)
( ) ( )xxxxxxH 9451260378367072
1 35799 +−+−=
( ) ( )94547253150630457720
1 24681010 −+−+−= xxxxxxH
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